Struggling with Equivalence relations I am currently studying for my exams this summer. I have a hard time getting to grips with the following question on discrete mathematics:

Consider the relationship $T$ between ordered pairs of natural numbers such that $(a, b)$ is related to $(c, d):$
$$[(a, b) T (c, d)] \iff ad = bc$$
Is T an equivalence relation?

 A: So hopefully from your discrete mathematics class, you learned that an equivilance relation $\sim$ (or T if you prefer, the symbol is irrelevant) for some set $A$ satisfies the following properties:


*

*Identity: $\forall a \in A$, $a \sim a$.

*Symmetry: $\forall a,b \in A$, if $a \sim b$ then $b \sim a$

*Transitivity: $\forall a,b,c \in A$, if $a \sim b, b \sim c$ then $a \sim c$


In your case, $A$ is the set of pairs of numbers $(a,b)$.
A: To prove a relation ~ is an equivalence relation, we need to prove:


*

*Reflexive: $a$~$a$

*Symmetric: $a$~$b$ implies $b$~$a$

*Transitive: $a$~$b$, $b$~$c$ implies $a$~$c$


If we define $(a,b)$~$(c,d) \iff ad=bc$, we then need to check all three properties above.
$(a,b)$~$(a,b)$ is true
$(a,b)$~$(c,d)\implies (c,d)$~$(a,b)$ is true
$(a,b)$~$(c,d)$, $(c,d)$~$(e,f)$ is not true, consider $(1,1),(0,0),(1,2)$
so it is not an equivalence relation on $\mathbb{N^+}\cup \{0\}$, as it is intransitive.
If however we only consider $\mathbb{N^+}$, it is an equivalence relation, as;
$\dfrac ab=\dfrac cd=\dfrac ef$.
